After a comprehensive study on the Montgomery modular inverse algorithm and its revised versions, two modified high radix algorithms are proposed which utilize higher radix to reduce iterations needed without increasing complexity much, thereby accelerating the process. The radix-4 algorithm can reduce the average number of iterations from 1.4n to 0.82n and a software experiment shows the speedup is about 11 percent and iterations are 41.5 percent less on average. The radix-8 algorithm can reduce the average number of iterations to 0.73n, but it is more complicated, which makes it suitable only for very large numbers (2,048 bits) in the experiment, where the speedup can be 13--18 percent. The proposed algorithms are suitable for software implementations on general-purpose microprocessors.